Middle East Technical University Spring 1994 Math 471 Elementary Topology and Geometry

Problem List 2 TOPOLOGICAL PROPERTIES AND CONSTRUCTIONS

Homeomorphisms

Problem 1. (2 pts) Prove that the complement of a closed interval in ℝ^{2} is homeomorphic to a punctured ℝ^{2}. Problem 2. (3 pts) Prove that two surfaces below are homeomorphic:

Product topology

Problem 3. (2 pts) Prove that any metric is continious as a map X × X → ℝ.

Quotient topology

Problem 4. (4 pts) Prove that the set of continious maps X/S → Y is in a 1–1 correspondence with the set of continious maps X → Y which are constant on the equivalence classes of S.

Separation properties

Problem 5. (2 pts) Prove that X is a Hausdorf topological space if and only if the diagonal in X × X is a closed subset.

Compactness

Problem 6. (1 pt) Prove that a locally bounded (not necessarily continious!) map X → ℝ is bounded for a compact X. Problem 7. (2 pts) Is it true that if F_{1} ⊂ F_{2} are two topologies in X, then compactness of (X, F_{1}) implies the compactness of (X, F_{2}) ? And vice versa ?

Connectedness

Problem 8. (1 pts) Prove that if A ⊂ X is a connected subset and A ⊂ B ⊂ Cl(A), then B is also connected. Problem 9. (2 pts) Prove that S^{1} is not homeomorphic to a subset of ℝ. Problem 10. (4 pts) Prove that if G is a topological group (a group with a topological structure such that the multiplication G × G → G is continious), then the connected component of the unit is a normal subgroup.

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